SLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

SLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.

K is INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation.

N

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

QSIZ

QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.

D

D is REAL array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.

Q

Q is REAL array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
on entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).

INDXQ

INDXQ is INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems
in D into ascending order. Note that elements in the second
half of this permutation must first have CUTPNT added to
their values in order to be accurate.

RHO

RHO is REAL
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
SLAED3.

CUTPNT

CUTPNT is INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.

Z

Z is REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.

DLAMDA

DLAMDA is REAL array, dimension (N)
A copy of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.

Q2

Q2 is REAL array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
SLAED7 in a matrix multiply (SGEMM) to update the new
eigenvectors.

LDQ2

LDQ2 is INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).

W

W is REAL array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to SLAED3.

PERM

PERM is INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.

GIVPTR

GIVPTR is INTEGER
The number of Givens rotations which took place in this
subproblem.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is REAL array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.

INDXP

INDXP is INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.

INDX

INDX is INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.